
where we have applied the formula that the lattice distortion \(U^{\parallel }_{t} = - U^{\parallel }_b\)  and are independent of the sublattice. In addition, the microscopic inter-layer hopping function are assumed to depend not only on the hopping displacement \(y + d_0 \hat{z}\) , but also on the direction of the nearest neighbor vectors \(n^{(\alpha )}_{t, S}\)  and \(n^{(\alpha )}_{b, S^{\prime }}\)  on each layer [1]}. Note that the terms in Eq. REF  are expanded only to the first order of the gradients of fermion fields, and therefore, the inter-layer tunneling can be written as
\(H_{inter} = \int {\rm d}^2 x\ \Psi _{t,S}(x) \left[ T_{SS^{\prime }}(x) + \frac{1}{2}\left\lbrace  p, \Lambda _{SS^{\prime }}(x) \right\rbrace  \right] \Psi _{b, S^{\prime }}(x) + h.c.\) 
